Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(25,26,31,34)(27,35,37,30)(28,38,40,32)(29,39,36,33);(1,2)(3,4)(5,7)(6,8)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,27)(26,30)(28,29)(31,37)(32,39)(33,38)(34,35)(36,40);(1,3,5,8)(2,6,7,4)(10,15)(11,17)(16,23)(18,24)(25,28,31,40)(26,32,34,38)(27,29,37,36)(30,39,35,33);(2,7)(3,8)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28)(26,32)(27,36)(29,37)(30,33)(31,40)(34,38)(35,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29,31,36)(26,33,34,39)(27,28,37,40)(30,38,35,32);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,30)(26,27)(28,39)(29,32)(31,35)(33,40)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8616425);